Ergodic process
In
Specific definitions
One can discuss the ergodicity of various statistics of a stochastic process. For example, a
and autocovariance
that depends only on the lag and not on time . The properties and are ensemble averages (calculated over all possible sample functions ), not
The process is said to be mean-ergodic[3] or mean-square ergodic in the first moment[4] if the time average estimate
Likewise, the process is said to be autocovariance-ergodic or d moment[4] if the time average estimate
converges in squared mean to the ensemble average , as . A process which is ergodic in the mean and autocovariance is sometimes called ergodic in the wide sense.
Discrete-time random processes
The notion of ergodicity also applies to discrete-time random processes for integer .
A discrete-time random process is ergodic in mean if
Examples
Ergodicity means the ensemble average equals the time average. Following are examples to illustrate this principle.
Call centre
Each operator in a call centre spends time alternately speaking and listening on the telephone, as well as taking breaks between calls. Each break and each call are of different length, as are the durations of each 'burst' of speaking and listening, and indeed so is the rapidity of speech at any given moment, which could each be modelled as a random process.
- Take N call centre operators (N should be a very large integer) and plot the number of words spoken per minute for each operator over a long period (several shifts). For each operator you will have a series of points, which could be joined with lines to create a 'waveform'.
- Calculate the average value of those points in the waveform; this gives you the time average.
- There are N waveforms and N operators. These N waveforms are known as an ensemble.
- Now take a particular instant of time in all those waveforms and find the average value of the number of words spoken per minute. That gives you the ensemble average for that instant.
- If ensemble average always equals time average, then the system is ergodic.
Electronics
Each resistor has an associated
Examples of non-ergodic random processes
- An unbiased random walk is non-ergodic. Its expectation value is zero at all times, whereas its time average is a random variable with divergent variance.
- Suppose that we have two coins: one coin is fair and the other has two heads. We choose (at random) one of the coins first, and then perform a sequence of independent tosses of our selected coin. Let X[n] denote the outcome of the nth toss, with 1 for heads and 0 for tails. Then the ensemble average is 1⁄2 (1⁄2 + 1) = 3⁄4; yet the long-term average is 1⁄2 for the fair coin and 1 for the two-headed coin. So the long term time-average is either 1/2 or 1. Hence, this random process is not ergodic in mean.
See also
- Ergodic hypothesis
- Ergodicity
- Ergodic theory, a branch of mathematics concerned with a more general formulation of ergodicity
- Loschmidt's paradox
- Poincaré recurrence theorem
Notes
References
- Porat, B. (1994). Digital Processing of Random Signals: Theory & Methods. Prentice Hall. p. 14. ISBN 0-13-063751-3.
- Papoulis, Athanasios (1991). Probability, random variables, and stochastic processes. New York: McGraw-Hill. pp. 427–442. ISBN 0-07-048477-5.